If you are planning to adopt a pet, then you'll need to pick out a vet anyway. So instead of waiting until your new pet is in your home, choose a vet and make an initial appointment now. (The shelter will probably expect you to have your new pet seen by a vet within a few weeks of the adoption anyway.) Or, if timing is too uncertain to schedule an appointment, at least call and ask what you need to do to register a new patient. You can then tell the shelter "I don't have a pet yet, but I will be seeing so-and-so" (with contact info).
The shelter I adopted from asked for the name of the vet I would be seeing, not for a reference. They don't expect people without pets to produce a reference, but they do expect you to have already made arrangements. I suspect your shelter doesn't really expect you to produce a veterinary reference; that seems unrealistic for first-time adopters, and shelters are constantly trying to reach more potential adopters. (Face it, there's a limit to the number of pets any one person can adopt, so while repeat clients are nice too, they have to broaden their search. And I note that mine offers classes for first-time pet owners.) If they do in fact require a veterinary reference, ask them what would satisfy them.